Methods for analysis of financial markets

ABSTRACT

A preferred embodiment comprises a method for obtaining predictive information (e.g., volatility) for inhomogeneous financial time series. Major steps of the method comprise the following: (1) financial market transaction data is electronically received by a computer over an electronic network; (2) the received financial market transaction data is electronically stored in a computer-readable medium accessible to the computer; (3) a time series z is constructed that models the received financial market transaction data; (4) an exponential moving average operator is constructed; (5) an iterated exponential moving average operator is constructed that is based on the exponential moving average operator; (6) a linear, time-translation-invariant, causal operator Ω[z] is constructed that is based on the iterated exponential moving average operator; (7) values of one or more predictive factors relating to the time series z and defined in terms of the operator Ω[z] are calculated by the computer; and (8) the values calculated by the computer are stored in a computer readable medium.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional ApplicationSer. No. 60/200,742, filed May 1, 2000; U.S. Provisional ApplicationSer. No. 60/200,743, filed May 1, 2000; U.S. Provisional ApplicationSer. No. 60/200,744, filed May 1, 2000; U.S. Provisional Application No.Ser. 60/261,973, filed Jan. 17, 2001; and U.S. Provisional ApplicationSer. No. 60/274,174, filed Mar. 8, 2001. The contents of the aboveapplications are incorporated herein in their entirety by reference.

FIELD OF THE INVENTION

[0002] The subject invention is related to inhomogeneous time seriesanalysis, and more particularly to the analysis of high-frequencyfinancial data, such as foreign exchange data.

BACKGROUND

[0003] Time series are the common mathematical framework used torepresent the world of economics and finance. Among time series, thefirst important classification can be done according to the spacing ofdata points in time. Regularly spaced time series are calledhomogeneous, irregularly spaced series are inhomogeneous. An example ofa homogeneous time series is a series of daily data, where the datapoints are separated by one day (on a business time scale, which omitsthe weekends and holidays) .

[0004] In most references on time series analysis, the time series to betreated are restricted to the filed of homogeneous time series (see,e.g., Granger C. W. J. and Newbold P., 1977, Forecasting economic timeseries, Academic Press, London; Priestley M. B., 1989, Non-linear andnon-stationary time series analysis, Academic Press, London; Hamilton J.D., 1994, Time Series Analysis, Princeton University Press, Princeton,N.J.) (hereinafter, respectively, Granger and Newbold, 1977; Priestley,1989; Hamilton, 1994). This restriction induces numeroussimplifications, both conceptually and computationally, and wasjustified before fast, inexpensive computers and high-frequency timeseries were available.

[0005] Current empirical research in finance is confronted with anever-increasing amount of data, caused in part by increased computerpower and communication speed. Many time series can be obtained at highfrequency, often at market tick-by-tick frequency. These time series areinhomogeneous, since market ticks arrive at random times. Inhomogeneoustime series by themselves are conceptually simple; the difficulty liesin efficiently extracting and computing information from them.

SUMMARY

[0006] There is thus a need for methods of analyzing inhomogeneous timeseries. Time series based on foreign exchange rates represent a standardexample of the practical application of such methods. In practice, themethods described herein are also suitable for applications tohomogeneous time series. Given a time series z, such as an asset price,the general point of view is to compute another time series, such as thevolatility of the asset, by the application of an operator Ω[z]. Thereis a need for a method of applying a set of basic operators that can becombined to compute more sophisticated quantities (for example,different kinds of volatility or correlation). In such a method, a fewimportant considerations must be kept in mind. First, the computationsmust be efficient. Even if powerful computers are becoming cheaper,typical tick-by-tick data in finance is 100 or even 10,000 times denserthan daily data. Clearly, one cannot afford to compute a fullconvolution for every tick. The basic workhorse is the exponentialmoving average (EMA) operator (described below), which can be computedvery efficiently through an iteration formula. A wealth of complex butstill efficient operators can be constructed by combining and iteratingthe basic operators described.

[0007] Second, stochastic behavior is the dominant characteristic offinancial processes. For tick-by-tick data, not only the values but alsothe time points of the series are stochastic. In this random world,point-wise values are of little significance and we are more interestedin average values inside intervals. Thus the usual notion of return alsohas to be changed. With daily data, a daily return is computed asr_(l)=p_(l)−p_(l−1), i.e., as a point-wise difference between the pricetoday and the price yesterday. With high-frequency data, a betterdefinition of the daily return is the difference between the averageprice of the last few hours and an average price from one day ago. Inthis way, it is possible to build smooth variables well-suited to randomprocesses. The calculus has to be revisited in order to replacepoint-wise values by averages over some time intervals.

[0008] Third, analyzing data typically involves a characteristic timerange; a return r[τ], for example, is computed on a given time intervalτ. With high-frequency data, this characteristic time interval can varyfrom few minutes to several weeks. We have been careful to make explicitall these time range dependencies in the formulation of operators usedin the described methods.

[0009] Finally, we often want smooth operators. Of course, there is asingularity at t=now, corresponding to the arrival of new information.This new information must be incorporated immediately, and therefore,the operators may have a jump behavior at t=now. Yet, aside from thisfundamental jump created by the advance of events, it is better to havecontinuous and smooth operators. A simple example of a discontinuousoperator is an average with a rectangular weighting function, say ofrange τ. The second discontinuity at now-τ, corresponding to forgettingevents, is unnecessary and creates spurious noise. Instead, a preferredembodiment uses moving average weighting functions (kernels) with asmooth decay to zero.

[0010] The above-listed goals are satisfied by the subject invention. Apreferred embodiment comprises a method to obtain predictive information(e.g., volatility) for inhomogeneous financial time series. Major stepsof the method comprise the following: (1) financial market transactiondata is electronically received by a computer over an electronicnetwork; (2) the received financial market transaction data iselectronically stored in a computer-readable medium accessible to thecomputer; (3) a time series z is constructed that models the receivedfinancial market transaction data; (4) an exponential moving averageoperator is constructed; (5) an iterated exponential moving averageoperator is constructed that is based on the exponential moving averageoperator; (6) a time-translation-invariant, causal operator Ω[z] isconstructed that is based on the iterated exponential moving averageoperator; (7) values of one or more predictive factors relating to thetime series z and defined in terms of the operator Ω[z] are calculatedby the computer; and (8) the values calculated by the computer arestored in a computer readable medium.

[0011] Various predictive factors are described below, and specificallycomprise return, momentum, and volatility. Other predictive factors willbe apparent to those skilled in the art.

[0012] The above briefly described embodiment is only one of severalpreferred embodiments described herein, and should not be interpreted asrepresenting the invention as a whole, or as the “thrust” of theinvention. Descriptions of other, equally important, embodiments havebeen omitted from this Summary merely for conciseness. Of particularnote is the fact that the described method is applicable to any timeseries data, not just FX data.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013]FIG. 1 is a graph of the foreign exchange rate for USD/CHF for theweek from Sunday, October 26, to Sunday, Nov. 2, 1997.

[0014]FIG. 2 is a graph of a kernel ma[τ, n](t) for n=1, 2, 4, 8, and16, where τ=1.

[0015]FIG. 3 is a graph on a logarithmic scale of the kernel ma[τ, n](t)for n=1, 2, 4, 8, and 16, where τ=1.

[0016]FIG. 4 is a graph of a schematic differential kernel.

[0017]FIG. 5 is a graph of a differential operator Δ[τ], for τ=1.

[0018]FIG. 6 is a graph on a logarithmic scale of the absolute value ofthe differential operator Δ[τ], for τ=1.

[0019]FIG. 7 illustrates a comparison between the differential computedusing the formula

Δ[τ]=γ (EMA[ατ, 1]+EMA[ατ, 2]−2 EMA[αβτ, 4]),

[0020] with τ=24 hours (“24 h”), and the point-wise return x(t)−x(t−24h).

[0021]FIG. 8 is a graph of an annualized derivative D[τ, γ=0.5; x] forUSD/CHF from 1 Jan. 1988 to 1 Nov. 1998.

[0022]FIG. 9 shows an annualized volatility computed as MNorm [τ/2; D[τ/32, γ=0.5; x]] with τ=1h.

[0023]FIG. 10 shows plots of a standardized return, a moving skewness,and a moving kurtosis.

[0024]FIG. 11 plots a kernel wf(t) for a windowed Fourier operator, forn=8 and k=6.

[0025]FIG. 12 shows a plot of a normed windowed Fourier transform forthe example week, with τ=1 hour, k=6, and n=8.

[0026]FIG. 13 illustrates major steps of a preferred embodiment.

[0027]FIG. 14 illustrates major steps of a second preferred embodiment.

[0028]FIG. 15 illustrates major steps of a third preferred embodiment.

[0029]FIG. 16 illustrates major steps of a fourth preferred embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0030] 1 Introduction

[0031] The generalization to inhomogeneous time series introduces anumber of technical peculiarities. Because of their time-translationinvariance, all macroscopic operators can be represented byconvolutions. A convolution is defined as an integral, so the seriesshould be defined in continuous time. Actual data is known only atdiscrete sampling times, so some interpolation needs to be used in orderto properly define the convolution integral. The same problem is presentwhen constructing an artificial homogeneous time series frominhomogeneous data. Another technical peculiarity originates from thefact that our macroscopic operators are ultimately composed of iteratedmoving averages. All such EMA operators have non-compact kernels: thekernels decay exponentially fast, but strictly speaking they arepositive. This implies an infinite memory; a build-up must be done overan initialization period before the value of an operator becomesmeaningful. All the above points are discussed in detail below.

[0032] High-frequency data in finance has a property that createsanother technical difficulty: strong intra-day and intra-weekseasonalities, due to the daily and weekly pattern of human activities.A powerful deseasonalization technique is needed, such as a transformedbusiness time scale (see Dacorogna, M. M., Müller, U. A., Nagler, R. J.,Olsen, R. B., and Pictet, O. V., 1993, A geographical model for thedaily and weekly seasonal volatility in the FX market, Journal ofInternational Money and Finance, 12(4), 413-438.) (hereinafter Dacorognaet al., 1993). Essentially, this scale is a continuous-timegeneralization of the familiar daily business time scale (which containsfive days per week, Saturdays and Sundays omitted). A continuousbusiness time scale θ allows us to map a time interval dt in physicaltime to an interval dθ in business time, where dθ/dt is proportional tothe expected market activity. All the techniques presented in this papercan be based on any business time scale. The required modification is toreplace physical time intervals with corresponding business timeintervals. As this extension is straightforward to those skilled in theart, all the formulae are given in physical time and a few remarks onscaled time are made when the extension or its consequences arenonobvious.

[0033] The plan of this description is as follows: The notation is fixedin Section 2 and the main theoretical considerations are given inSection 3. A set of convenient macroscopic operators, includingdifferent moving averages and derivatives are given in Section 4. Armedwith powerful basic operators, it is then easy to introduce novelmethods of calculating time series predictive factors such as movingvolatility, correlation, moving skewness and kurtosis, and to generalizethe described methods to complex-valued operators. In Section 5, wedescribe preferred implementations of the method.

[0034] Examples are given with data taken from the foreign exchange (FX)market. When not specified, the data set is USD/CHF for the week fromSunday, Oct. 26, 1997 to Sunday, November 2. This week has been selectedbecause on Tuesday, Oct. 28, 1997, some Asian stock markets crashed,causing turbulences in many markets around the world, including the FXmarket. Yet, the relation between a stock market crash originating inAsia and the USD/CHF foreign exchange rate is quite indirect, makingthis example interesting. The prices of USD/CHF for the example week areplotted in FIG. 1. All the figures for this week have been computedusing high-frequency data; the results have finally been sampled eachhour using a linear interpolation scheme. The computations have beendone in physical time, therefore exhibiting the full daily and weeklyseasonalities contained in the data. FIG. 1 shows the FX rate of USD/CHF(U.S. dollars to Swiss Francs) for the week of Sunday, October 26 toSunday, Nov. 2, 1997. On the time axis, the labels correspond to the dayin October, with the points 32 and 33 corresponding to November 1 and 2.From the market quote containing bid and ask prices, the (geometric)middle price was computed as {square root}{square root over (bid·ask)}.

[0035] Finally, we want to emphasize that the techniques describedherein can be applied to a wide range of statistical computations infinance—for example, the analysis needed in risk management. Awell-known application can be found in (Pictet O. V., Dacorogna M. M.,Müller U. A., Olsen R. B.,, and Ward J. R., 1992, Real-time tradingmodels for foreign exchange rates, Neural Network World, 2(6), 713-744.)(hereinafter Pictet et al., 1992). Further, these techniques can also beapplied to any time series data (e.g., commodity prices or temperaturedata), not just financial data.

[0036] 2 Notation and Mathematical Preliminaries

[0037] The letter z is used herein to represent a generic time series.The elements, or ticks, (t_(l), z_(i)) of a time series z consist of atime t_(l) and a scalar value z_(l). The generalization to multivariateinhomogeneous time series is straightforward (except for the businesstime scale aspect). The value z_(i)=z(t_(l)) and the time point t_(i)constitute the i-th element of the time series z. The sequence ofsampling (or arrival) times is required to be growing: t_(i)>t_(l−1).The strict inequality is required in a true univariate time series andis theoretically always true if the information arrives through onechannel. In practice, the arrival time is only known with a finiteprecision (say, one second), and two ticks may well have the samearrival time. Yet, for most of the methods described herein, the strictmonotonicity of the time process is not required. A general time seriesis inhomogeneous, meaning that the sampling times are irregular. For anhomogeneous time series, the sampling times are regularly spaced:t_(l)−t_(l−1)=δt=constant. If a time series depends on some parametersθ, these are made explicit between square brackets, z[θ].

[0038] An operator Ω from the space of time series into itself isdenoted by Ω[z]. The operator may depend on some parameters Ω[θ;z]. Thevalue of Ω[z] at time t is Ω[z](t). For linear operators, a productnotation Ωz is also used. The average over a whole time series of lengthT is denoted by E[z]:=1T ∫dt z(t). The probability density function(pdf) of z is denoted p(z). A synthetic regular (or homogeneous) timeseries (RTS), spaced by δt, derived from the irregular time series z, isnoted RTS[δt;z]. A standardized time series for z is denoted {circumflexover (z)}=(z−E[z])/σ[z], where σ[z]²=E[(z−E[z])²].

[0039] The letter x is used to represent the logarithmic middle pricetime series x=(Inp_(bid)+Inp_(ask))/2=In{square root}{square root over(p_(bid)p_(ask))}.

[0040] 3 Convolution Operators: General Considerations

[0041] 3.1 Linear Operators

[0042] If an operator is linear, time-translation invariant and causal,it can be represented by a convolution with a kernel ω(t):$\begin{matrix}\begin{matrix}{{{\Omega \lbrack z\rbrack}(t)} = \quad {\int_{- \infty}^{t}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}{z\left( t^{\prime} \right)}}}} \\{= \quad {\int_{0}^{\infty}\quad {{t^{\prime}}{\omega \left( t^{\prime} \right)}{{z\left( {t - t^{\prime}} \right)}.}}}}\end{matrix} & (1)\end{matrix}$

[0043] The kernel ω(t) is defined only on the positive semi-axis t≧0,and should decay to t large enough. With this convention for theconvolution, the weight given to past events corresponds to the value ofthe kernel for positive argument. The value of the kernel ω(t−t′) is theweight of events in the past, at a time interval t−t′ from t. In thisconvolution, z(t^(l)) is a continuous function of time. Actual timeseries z are known only at the sampling time t_(l) and should beinterpolated between sampling points. Many interpolation procedures forthe value of z(t) between t_(l−1) and t_(l) can be defined, but threeare used in practice: previous value z(t)=z_(i−1), next valuez(t)=z_(l), and linear interpolation z(t)=α(t)z_(l−1)+[1−α(t)]z_(i) withα(t)=(t_(l)−t)/(t_(i)−t_(i−1)).

[0044] The linear interpolation leads to a continuous interpolatedfunction. Moreover, linear interpolation defines the mean path of arandom walk, given the start and end values. Unfortunately, it isnon-causal, because in the interval between t_(l−1) and t_(l), the valueat the end of the interval z_(l) is used. Only the previous-valueinterpolation is causal, as only the information known at t_(i−1) isused in the interval between t_(l−1) and t_(l). Any interpolation can beused for historical computations, but for the real-time situation, onlythe causal previous-value interpolation is defined. In practice, theinterpolation scheme is almost irrelevant for good macroscopicoperators—i.e., if the kernel has a range longer than the typicalsampling rate.

[0045] The kernel ω(t) can be extended to all t∈R, with ω(t)=0 for t<0.This is useful for analytical computation, particularly when the orderof integral evaluations has to be changed. If the operator Ω is linearand time-translation invariant but non-causal, the same representationcan be used except that the kernel may be non-zero on the whole timeaxis.

[0046] Two broad families of operators that share general shapes andproperties are often used. An average operator has a kernel that isnon-negative, ω(t)≧0, and normalized to unity, ∫dt ω(t)=1. This impliesthat Ω[parameters; Const]=Const. Derivative and difference operatorshave kernels that measure the difference between a value now and a valuein the past (with a typical lag of τ). Their kernels have a zero average∫dt ω(t)=0, such that Ω[parameters;Const]=0.

[0047] The integral (1) can also be evaluated in scaled time. In thiscase, the kernel is no more invariant with respect to physical timetranslation (i.e., it depends on t and t′) but it is invariant withrespect to translation in business time. If the operator is an averageor a derivative, the normalization property is preserved in scaled time.

[0048] 3.2 Range and Width

[0049] The nth moment of a causal kernel ω is defined as $\begin{matrix}{{\langle t^{n}\rangle}_{\omega} = {\int_{0}^{\infty}\quad {{t^{\quad}}{\omega \left( t^{\quad} \right)}{t^{n}.}}}} & (2)\end{matrix}$

[0050] The range r and the width w of an operator Ω are definedrespectively by the following relations $\begin{matrix}\begin{matrix}{{{r\lbrack\Omega\rbrack} = \quad {{\langle t^{\quad}\rangle}_{\omega} = {\int_{0}^{\infty}\quad {{t^{\quad}}{\omega \left( t^{\quad} \right)}t^{\quad}}}}},} \\{{w^{2}\lbrack\Omega\rbrack} = \quad {{\langle\left( {t - r} \right)^{2}\rangle}_{\omega} = {\int_{0}^{\infty}\quad {{t^{\quad}}{\omega \left( t^{\quad} \right)}{\left( {t - r} \right)^{2}.}}}}}\end{matrix} & (3)\end{matrix}$

[0051] For most operators Ω[τ] depending on a time range τ, the formulais set up so that |r[Ω[τ]]|=τ.

[0052] 3.3 Convolution of Kernels

[0053] A standard step is to successively apply two linear operators:

Ω_(c) [z]=Ω ₂ oΩ ₁ [z]=Ω ₂Ω₁ z:=Ω ₂[Ω₁ [z]].

[0054] It is easy to show that the kernel of Ω_(c) is given by theconvolution of the kernels of Ω1 and Ω₂: $\begin{matrix}{\omega_{c} = {{\omega_{1}\quad \bigstar \quad \omega_{2}\quad {or}\quad {\omega_{c}\left( {t - t^{\prime}} \right)}} = {\int_{- \infty}^{\infty}\quad {{t^{''}}{\omega_{1}\left( {t - t^{''}} \right)}{\omega_{2}\left( {t^{''} - t^{\prime}} \right)}}}}} & (4)\end{matrix}$

[0055] or, for causal operators, $\begin{matrix}{{{\omega_{c}(t)} = {{\int_{{- t}/2}^{t/2}\quad {{t^{\prime}}{\omega_{1}\left( {\frac{t}{2} - t^{\prime}} \right)}{\omega_{2}\left( {t^{\prime} + \frac{t}{2}} \right)}\quad {for}\quad t}} \geq 0}},} & (5)\end{matrix}$

[0056] and Ω_(c)(t)=0 for t<0. Under convolution, range, width, andsecond moment obey the following simple laws: $\begin{matrix}\begin{matrix}{{r_{c} = {r_{1} + r_{2}}},} \\{{w_{c}^{2} = {w_{1}^{2} + w_{2}^{2}}},} \\{{\langle t^{\quad 2}\rangle}_{c} = {{\langle t^{\quad 2}\rangle}_{1} + {\langle t^{\quad 2}\rangle}_{2} + {2r_{1}{r_{2}.}}}}\end{matrix} & (6)\end{matrix}$

[0057] 3.4 Build-up Time Interval

[0058] Since the basic building blocks of a preferred embodiment are EMAoperators, most kernels have an exponential tail for large t. Thisimplies that, when starting the evaluation of an operator at time T, abuild-up time interval must elapse before the result of the evaluationis “meaningful,” (i.e., the initial conditions at T are sufficientlyforgotten). This heuristic statement can be expressed by quantitativedefinitions. We assume that the process z(t) is known since time T, andis modeled before T as an unknown random walk with no drift. Thedefinition (1) for an operator Ω computed since T needs to be modifiedin the following way: $\begin{matrix}{{{\Omega \left\lbrack {T;z} \right\rbrack}(t)} = {\int_{T}^{t}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}{{z\left( t^{\prime} \right)}.}}}} & (7)\end{matrix}$

[0059] The “infinite” build-up corresponds to Ω[−∞; z](t). For −T<0, theaverage build-up error ε at t=0 is given by $\begin{matrix}{ɛ^{2} = {{E\left\lbrack \left( {{{\Omega \left\lbrack {{- T};z} \right\rbrack}(0)} - {{\Omega \left\lbrack {{- \infty};z} \right\rbrack}(0)}} \right)^{2} \right\rbrack} = {E\left\lbrack \left( {\int_{- \infty}^{- T}\quad {{t^{\prime}}{\omega \left( {- t^{\prime}} \right)}{z\left( t^{\prime} \right)}}} \right)^{2} \right\rbrack}}} & (8)\end{matrix}$

[0060] where the expectation E[ ] is an average on the space ofprocesses z. For a given build-up error ε, this equation is the implicitdefinition of the build-up time interval T. In order to compute theexpectation, we need to specify the considered space of randomprocesses. We assume simple random walks with constant volatility σ,namely $\begin{matrix}{{E\left\lbrack \left( {{z(t)} - {z\left( {t + {\delta t}} \right)}} \right)^{2} \right\rbrack} = {\sigma {\frac{\delta \quad t}{1y}.}}} & (9)\end{matrix}$

[0061] The symbol 1y denotes one year, so δt/1y is the length of δtexpressed in years. With this choice of units, σ is an annualizedvolatility, with values roughly from 1% (for bonds) to 50% (for stocks),and a typical value of 10% for foreign exchange. For t<−T, t′<−T, wehave the expectation $\begin{matrix}{{E\left\lbrack {{z(t)}{z\left( t^{\prime} \right)}} \right\rbrack} = {{z\left( {- T} \right)}^{2} + {\sigma \quad {{\min \left( {\frac{{- t} - T}{1y},\frac{{- t^{\prime}} - T}{1y}} \right)}.}}}} & (10)\end{matrix}$

[0062] Having defined the space of processes, a short computation gives$\begin{matrix}{ɛ^{2} = {{{z\left( {- T} \right)}^{2}\left( {\int_{t}^{\infty}\quad {{t}\quad {\Omega (t)}}} \right)^{2}} + {2\quad \sigma {\int_{T}^{\infty}\quad {{t}\quad {\omega (t)}{\int_{T}^{t}\quad {{t^{\prime}}{\omega \left( t^{\prime} \right)}{\frac{t^{\prime} - T}{1y}.}}}}}}}} & (11)\end{matrix}$

[0063] The first term is the “error at initialization,” corresponding tothe decay of the initial value Ω[−T](−T)=0 in the definition (7). Abetter initialization is Ω[−T](−T)=z(−T)∫₀ ^(∞)ω(t), corresponding to amodified definition for Ω[T](t): $\begin{matrix}{{{\Omega \left\lbrack {T;z} \right\rbrack}(t)} = {{{z\left( {- T} \right)}{\int_{- \infty}^{T}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}}}} + {\int_{T}^{t}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}{{z\left( t^{\prime} \right)}.}}}}} & (12)\end{matrix}$

[0064] Another interpretation for the above formula is that z isapproximated by its most probable value z(−T) for t<T. With this betterdefinition for Ω, the error reduces to $\begin{matrix}{ɛ^{2} = {2\quad \sigma {\int_{T}^{\infty}\quad {{t}\quad {\omega (t)}{\int_{T}^{t}\quad {{t^{\prime}}{\omega \left( t^{\prime} \right)}{\frac{t^{\prime} - T}{1y}.}}}}}}} & (13)\end{matrix}$

[0065] For a given kernel ω, volatility σ and error ε, eq. (13) is anequation for T. Most of the kernels introduced in the next section havethe scaling form ω(τ,t)={tilde over (ω)}(t/τ)/τ. In this case, theequation for $\overset{\sim}{T} = \frac{T}{\tau}$

[0066] reduces to $\begin{matrix}{ɛ^{2} = {2\quad \sigma \frac{\tau}{1y}{\int_{\overset{\sim}{T}}^{\infty}\quad {{t^{\quad}}{\overset{\sim}{\omega}\left( t^{\quad} \right)}{\int_{\overset{\sim}{T}}^{t}\quad {{t^{\prime}}{\overset{\sim}{\omega}\left( t^{\prime} \right)}{\left( {t^{\prime} - \overset{\sim}{T}} \right).}}}}}}} & (14)\end{matrix}$

[0067] Since this equation cannot be solved for general operators, thebuild-up interval should be computed numerically. This equation can besolved analytically for the simple EMA kernel, and gives the solutionfor the build-up time: $\begin{matrix}{\frac{T}{\tau} = {{{- \ln}\quad ɛ} + {\frac{1}{2}{{\ln \left( {\frac{\sigma}{2}\frac{\tau}{1y}} \right)}.}}}} & (15)\end{matrix}$

[0068] As expected, the build-up time interval is large for a smallerror tolerance and for processes with high volatility.

[0069] For operators more complicated than the simple EMA, eq. (14) is,in general, not solvable analytically. A simple rule of thumb can begiven: the fatter the tail of the kernel, the longer the requiredbuild-up. A simple measure for the tail can be constructed from thefirst two moments of the kernel as defined by eq. (2). The aspect ratioAR[Ω] is defined as $\begin{matrix}{{A\quad {R\lbrack\Omega\rbrack}} = \frac{{\langle t^{2}\rangle}_{\omega}^{1/2}}{{\langle t^{\quad}\rangle}_{\omega}}} & (16)\end{matrix}$

[0070] Both (t) and {square root}{square root over (<t²>)} measure theextension of the kernel and are usually proportional to τ; thus theaspect ratio is independent of τ (the “width” of the moving “window” ofdata over which the EMA is “averaged”) and dependent only on the shapeof the kernel, in particular its tail property. Typical values of thisaspect ratio are 2/{square root}{square root over (3)} for a rectangularkernel and {square root}{square root over (2)} for a simple EMA. A lowaspect ratio means that the kernel of the operator has a short tail andtherefore a short build-up time interval in terms of τ. This is a goodrule for non-negative causal kernels; the aspect ratio is less usefulfor choosing the build-up interval of causal kernels with morecomplicated, partially negative shapes.

[0071] 3.5 Homogeneous Operators

[0072] There are many more ways to build non-linear operators; anexample is given in Section 4.8 for the (moving) correlation. Inpractice, most non-linear operators are homogeneous of degree p, namelyΩ[ax]=|a|^(p)Ω[x] (here the word “homogeneous” is used in a sensedifferent from that in the term “homogeneous time series”).Translation-invariant homogeneous operators of degree pq take the simpleform of a convolution: $\begin{matrix}{{{\Omega \lbrack z\rbrack}(t)} = \left\lbrack {\int_{- \infty}^{t}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}{{z\left( t^{\prime} \right)}}^{p}}} \right\rbrack^{q}} & (17)\end{matrix}$

[0073] for some exponents p and q. An example is the moving norm (seeSection 4.4) with ω corresponding to an average and q=1/p.

[0074] 3.6 Robustness

[0075] Data errors (outliers) should be filtered prior to anycomputation. Outlier filtering is difficult and sometimes arbitrary forhigh-frequency data in finance; this data is stochastic with afat-tailed distribution of price changes (see Pictet O. V., Dacorogna M.M., and Müller U. A., Hill, bootstrap and jackknife estimators for heavytails, in “A practical guide to heavy tails: Statistical Techniques forAnalysing Heavy Tailed Distributions,” edited by Robert J. Adler, RaisaE. Feldman and Murad S. Taqqu, published by Birkhauser, Boston 1998)(hereinafter Pictet et al., 1998). Sometimes it is desirable to buildrobust estimators to reduce the impact of outliers and the choice of thefiltering algorithm. The problem is acute mainly when working withreturns, for example when estimating a volatility, because thedifference operator needed to compute the return r from the price x issensitive to outliers. The following modified operator achievesrobustness by giving a higher weight to the center of the distributionof returns r than to the tails:

Ω[f;r]=f ⁻¹ {Ω[f(r)]}  (18)

[0076] where f is an odd function over R. Possible mapping functionsf(x) are

sign (x)|x| ^(γ) =x|x| ^(γ−1),  (19)

sign(x) (this corresponds to γ→0 in the above formula),  (20)

tan h(x/x₀).  (21)

[0077] Robust operator mapping functions defined by eq. (19) have anexponent 0≦γ<1. In some special applications, operators with γ>1,emphasizing the tail of the distribution, may also be used. In thecontext of volatility estimates, the usual L² volatility operator basedon squared returns can be made more robust by using the mapping functionf=sign (x) {square root}{square root over (|x|)} (the signed squareroot); the resulting volatility is then based on absolute returns as ineq. (39). More generally, the signed power f(x)=sign(x)|x|^(p)transforms an L² volatility into an L^(2p) volatility. This simple powerlaw transformation is often used and therefore included in thedefinition of the moving norm, moving variance, or volatility operators,eq. (32). Yet, as will be apparent to those skilled in the art, moregeneral transformations can also be used.

[0078] 4 The Menagerie of Convolution Operators

[0079] 4.1 Exponential moving average EMA[τ]

[0080] The basic exponential moving average (EMA) is a simple averageoperator, with an exponentially decaying kernel: $\begin{matrix}{{{ema}(t)} = {\frac{e^{{- t}/\tau}}{\tau}.}} & (22)\end{matrix}$

[0081] This EMA operator is our foundation. Its computation is veryefficient, and other more complex operators can be built with it, suchas MAs, differentials, derivatives and volatilities. The numericalevaluation is efficient because of the exponential form of the kernel,which leads to a simple iterative formula: $\begin{matrix}\begin{matrix}{{{{EMA}\left\lbrack {\tau;z} \right\rbrack}\left( t_{n} \right)} = \quad {{\mu \quad {{EMA}\left( {\tau;z} \right\rbrack}\left( t_{n - 1} \right)} +}} \\{\quad {{{\left( {v - \mu} \right)z_{n - 1}} + {\left( {1 - v} \right)z_{n}}},{with}}} \\{{\alpha = \quad \frac{\tau}{t_{n} - t_{n - 1}}},} \\{{\mu = \quad e^{- \alpha}},}\end{matrix} & (23)\end{matrix}$

[0082] and where v depends on the chosen interpolation scheme,$\begin{matrix}{v = \left\{ \begin{matrix}1 & {{previous}\quad {point}} \\{\left( {1 - \mu} \right)/\alpha} & {{linear}\quad {interpolation}} \\\mu & {{next}\quad {point}}\end{matrix} \right.} & (24)\end{matrix}$

[0083] Thanks to this iterative formula, the convolution never needs tobe computed in practice; only a few multiplications and additions haveto be done for each tick. In section 4.10, the EMA operator is extendedto the case of complex kernels.

[0084] 4.2 The Iterated EMA [τ, n]

[0085] The basic EMA operator can be iterated to provide a family ofiterated exponential moving average operators EMA [τ, n]. A simplerecursive definition is

EMA[τ,n; z]=EMA[τ; EMA[τ, n−1; z]]  (25)

[0086] with EMA[τ, 1; z]=EMA [τ, z]. This definition can be efficientlyevaluated by using the iterative formula (23) for all these basic EMAs.There is a non-obvious complication related to the choice of theinterpolation scheme (24). The EMA of z necessarily has an interpolationscheme different from that used for z. The correct form of EMA [τ; z]between two points is no longer a straight line but a non-linear(exponential) curve. It will be straightforward to those skilled in theart to derive the corresponding exact interpolation formula. When one ofthe interpolation schemes of eq. (24) is used after the first iteration,a small error is made. Yet, if the kernel is wide as compared to tot_(n)−t_(n−1), this error is indeed very small. As a suitableapproximation, a preferred embodiment uses linear interpolation in thesecond and all further EMA iterations, even if the first iteration wasbased on the next-point interpolation. The only exception occurs if z,is not yet known; then we need a causal operator based on theprevious-point interpolation.

[0087] The kernel of EMA[τ, n] is $\begin{matrix}{{{{ema}\left\lbrack {\tau,n} \right\rbrack}(t)} = {\frac{1}{\left( {n - 1} \right)!}\left( \frac{t}{\tau} \right)^{n - 1}{\frac{e^{{- t}/\tau}}{\tau}.}}} & (26)\end{matrix}$

[0088] This family of functions is related to Laguerre polynomials,which are orthogonal with respect to the measure e^(−l) (for τ=1).Through an expansion in Laguerre polynomials, any kernel can beexpressed as a sum of iterated EMA kernels. Therefore, the convolutionwith an arbitrary kernel can be evaluated by iterated exponential movingaverages. Yet, the convergence of this expansion may be slow, namelyhigh-order iterated EMAs may be necessary, possibly with very largecoefficients. This typically happens if one tries to construct operatorsthat have a decay other (faster) than exponential. Therefore, inpractice, we construct operators “empirically” from a few low-orderEMAs, in a way to minimize the build-up time. The set of operatorsprovided by this description covers a wide range of computations neededin finance.

[0089] The range, second moment, width, and aspect ratio of the iteratedEMA are, respectively, $\begin{matrix}\begin{matrix}{{r = \quad {n\quad \tau}},} \\{{{\langle t^{2}\rangle} = \quad {{n\left( {n + 1} \right)}\tau^{2}}},} \\{{w^{2} = \quad {n\quad \tau^{2}}},} \\{{A\quad R} = \quad {\sqrt{\left( {n + 1} \right)n}.}}\end{matrix} & (27)\end{matrix}$

[0090] The iterated EMA[τ, n] operators with large n have a shorter,more compact kernel and require a shorter build-up time interval than asimple EMA of the same range nτ. This is indicated by the fact that theaspect ratio AR decreases toward 1 for large n. Each basic EMA operatorthat is part of the iterated EMA has a range τ which is much shorterthan the range nτ of the full kernel. Even if the tail of the kernel isstill exponential, it decays faster due to the small basic EMA range τ.

[0091] In order to further improve our preferred method, we buildanother type of compact kernel by combining iterated EMAs, as shown inthe next section. As the iterated EMAs, these combined iterated EMAshave a shorter build-up time interval than a simple EMA of the samerange.

[0092] 4.3 Moving Average MA[τ, n]

[0093] A very convenient moving average operator is provided by$\begin{matrix}{{{M\quad {A\left\lbrack {\tau,n} \right\rbrack}} = {\frac{1}{n}{\sum\limits_{k = 1}^{n}{E\quad M\quad {A\left\lbrack {\tau^{\prime},k} \right\rbrack}}}}},{{{wi}\quad t\quad h\quad \tau^{\prime}} = {\frac{2\quad \tau}{n + 1}.}}} & (28)\end{matrix}$

[0094] The parameter τ′ is chosen so that the range of MA[τ, n] is r=τ,independent of n. This provides a family of more rectangular-shapedkernels, with the relative weight of the distant past controlled by n.Kernels for different values of n and τ=1 are shown in FIG. 2, wherema[τ,n](t)is plotted for n=1, 2, 4, 8, and 16, with τ=1. The kernels'analytical form is given by $\begin{matrix}{{m\quad {a\left\lbrack {\tau,n} \right\rbrack}(t)} = {\frac{n + 1}{n}\frac{e^{{- t}/\tau^{\prime}}}{2\quad \tau}{\sum\limits_{k = 0}^{n - 1}{\frac{1}{k!}{\left( \frac{t}{\tau^{\prime}} \right)^{k}.}}}}} & (29)\end{matrix}$

[0095] For n=∞, the sum corresponds to the Taylor expansion ofexp(t/τ′), which cancels the term exp (−t/τ′) in (29), making the kernelconstant. For finite n, when t/τ′ is small enough, the finite sum willbe a very good approximation of exp (−t/τ′). “Small enough” means thatthe largest term in the sum is of order one: (t/τ′)^(n)/n!˜1. For largen, the condition (t/τ′)n/n!˜1 corresponds to t˜2τ (using Stirling'sapproximation n!˜n^(n)). Therefore, for t<<2τ, the series approximateswell the Taylor expansion of an exponential:$\left. {\sum\limits_{k = 0}^{n - 1}{\frac{1}{k!}\left( \frac{t}{\tau^{\prime}} \right)^{k}}}\rightarrow e^{t/\tau^{\prime}} \right.,\left. {m\quad a}\rightarrow{\frac{n + 1}{n}{\frac{1}{2\tau}.}} \right.$

[0096] This explains the constant behavior of the kernel for t<<2τ. Fort>2τ large, the exponential always dominates and the kernel decays tozero. Therefore, for large n, this operator tends to a rectangularmoving average for which AR=2/{square root}{square root over (3)}. Forvalues of n≧5, the kernel is rectangular-like more than EMA-like; thiscan as seen in FIG. 2. The decay of MA kernels is shown in FIG. 3, wherema[τ, n](t) is plotted on a logarithmic scale, for n=1, 2, 4, 8, and 16,with τ=1. The aspect ratio of the MA operator is $\begin{matrix}{{A\quad R} = {\sqrt{\frac{4\left( {n + 2} \right)}{3\left( {n + 1} \right)}}.}} & (30)\end{matrix}$

[0097] Clearly, the larger n, the shorter the build-up.

[0098] This family of operators can be extended by “peeling off” someEMAs with small k:${{MA}\left\lbrack {\tau,n_{\inf},n_{\sup}} \right\rbrack} = {{\frac{1}{n_{\sup} - n_{\inf} + 1}{\sum\limits_{k = n_{\inf}}^{n_{\sup}}{{{EMA}\left\lbrack {\tau^{\prime},k} \right\rbrack}\quad {with}\quad \tau^{\prime}}}} = \frac{2\quad \tau}{n_{\sup} + n_{\inf}}}$

[0099] and with 1≦n_(inf)≦n_(sup). By choosing such a modified MA withn_(inf)>1, we can generate a lag operator with a kernel whoserectangular-like form starts after a lag rather than immediately. Thisis useful in many applications that will be clear to those skilled inthe art.

[0100] In almost every case, a moving average operator can be usedinstead of a sample average. The sample average of z(t) is defined by$\begin{matrix}{{E\lbrack z\rbrack} = {\frac{1}{t_{e} - t_{s}}{\int_{t_{s}}^{t_{e}}\quad {{t^{\prime}}{z\left( t^{\prime} \right)}}}}} & (31)\end{matrix}$

[0101] where the dependency on start-time t_(S) and end-time _(the) isimplicit on the left-hand side. This dependency can be made explicitwith the notation E[t_(e)−t_(s); Z](_(the)), thus demonstrating theparallelism between the sample average and a moving average MA[2τ;z](t). The conceptual difference is that when using a sample average,t_(s) and _(the) are fixed, and the sample average is a functional fromthe space of time series to R, whereas the MA operator produces anothertime series. Keeping this difference in mind, we can replace the sampleaverage Et[·] by a moving average MA[·]. For example, we can construct astandardized time series z (as defined in Section 2), a moving skewness,or a moving correlation (see the various definitions below). Yet, sampleaverages and MAs can behave differently: for example,E[(z−E[z]²]=E[z²]−E[z]², whereas MA [(z−MA [z])²]≠MA [z²]−MA [z]².

[0102] 4.4 Moving Norm, Variance and Standard Deviation

[0103] With the efficient moving average operator, we define the movingnorm, moving variance, and moving standard deviation operators,respectively:

MNorm[τ, p; z]=MA[τ;|z|^(p)]^(1/p),

MVar[τ, p; z]=MA[τ;|z−MA[τ, z]|^(p)],

MSD[τ, p; z]=MA[τ;|z−MA[τ; z]|^(p)]^(1/p).  (32)

[0104] The norm and standard deviation are homogeneous of degree onewith respect to z. The p-moment up is related to the norm by μ_(p)=MA[|z|^(p)=mNORM[z]^(p). Usually, p=2 is taken. Lower values forp providea more robust estimate (see Section 3.6), and p=1 is another commonchoice. Even lower values can be used, for example p={fraction (1/2)}.In the formulae for MVar and MSD, there are two MA operators with thesame range τ and the same kernel. This choice is in line with standardpractice: empirical means and variances are computed for the samesample. Other choices can be interesting—for example, the sample menucan be estimated with a longer time range.

[0105] 4.5 Differential Δ[τ]

[0106] As mentioned above, a low-noise differential operator suitable tostochastic processes should compute an “average differential,” namely,the difference between an average around time “now” over a time intervalτ₁ and an average around time “now−τ” on a time interval τ₂. The kernelmay look like the schematic differential kernel plotted in FIG. 4.

[0107] Usually, τ, τ₁, and τ₂ are related and only the τ parameterappears, with τ₁˜τ₂˜τ/2. The normalization for Δ is chosen so that Δ[τ;c]=0 for a constant function c=c(t)=constant, and Δ[τ; t]=τ. Note thatour point of view is different from that used in continuous-timestochastic analysis. In continuous time, the limit τ→0 is taken, leadingto the Ito derivative with its subtleties. In our case, we keep therange τ finite in order to be able to analyze a process at differenttime scales (i.e., for different orders of magnitudes of τ). Moreover,for financial data, the limit τ→0 cannot be taken because a process isknown only on a discrete set of time points (and probably does not existin continuous time).

[0108] The following operator can be selected as a suitable differentialoperator:

Δ[τ]=γ(EMA[ατ, 1]+EMA[ατ, 2]−2EMA[αβτ, 4])  (33)

[0109] with γ=1.22208, β=0.65 and α⁻¹=γ(8β−3). This operator has awell-behaved kernel that is plotted in FIG. 5, wherein the full line 510is the graph of the kernel of the differential operator Δ[τ], for τ=1;the dotted curve 520 corresponds to the first two terms γ(EMA[Δτ,1]+EMA[ατ, 2]); and the dashed curve 530 corresponds to the last term2γEMA[αβτ, 4]. The value of γ is fixed so that the integral of thekernel from the origin to the first zero is one. The value of α is fixedby the normalization condition and the value of β is chosen in order toget a short tail.

[0110] The tail can be seen in FIG. 6, which shows the kernel of thedifferential operator Δ[τ], plotted in a logarithmic scale. The dottedline 610 shows a simple EMA with range τ, demonstrating the much fasterdecay of the differential kernel. After t=3.25τ, the kernel is smallerthan 10⁻³, which translates into a small required build-up time of about4τ.

[0111] In finance, the main purpose of a Δ operator is computing returnsof a time series of (logarithmic) prices x with a given time interval τ.Returns are normally defined as changes of x over τ; the alternativereturn definition r[τ]=Δ[τ; x] is used herein. This computation requiresthe evaluation of 6 EMAs and is therefore efficient, time-wise andmemory-wise. An example using our “standard week” is plotted in FIG. 7,demonstrating the low noise level of the differential. FIG. 7illustrates a comparison between the differential computed using theformula (33) with τ=24 hours (“24 h”) (the solid line 710), and thepoint-wise return x(t)−x(t−24 h) (the dotted line 720). The time lag ofapproximately 4 hours between the curves is essentially due to theextent of both the positive part of the kernel (0<t<0.5) and the tail ofthe negative part (t>1.5).

[0112] The conventionally computed return r[−τ](t)=x(t)−x(t−τ) is veryinefficient to evaluate for inhomogeneous time series. The computationof x(t - r) requires many old t_(i), x_(i) values to be kept in memory,and the t_(l) interval bracketing the time t−τ has to be searched for.Moreover, the number of ticks to be kept in memory is not bounded. Thisreturn definition corresponds to a differential operator kernel made oftwo δ functions (or to the limit τ₁, τ₂→0 of the kernel in FIG. 4). Thequantity x(t)−x(t−τ) can be quite noisy, so a further EMA might be takento smooth it. In this case, the resulting effective differentialoperator kernel has two discontinuities, at 0 and at τ, and decaysexponentially (much slower than the kernel of Δ[τ;x]). Thus it iscleaner and more efficient to compute returns with the Δ operator of eq.(33).

[0113] Another quantity commonly used in finance is x−EMA[τ;x], oftencalled a momentum or an oscillator. This is also a differential with thekernel δ(t)−exp(−t/τ)τ, with a δ function at t=0. A simple drawing showsthat the kernel of eq. (33) produces a much less noisy differential.Other appropriate kernels can be designed, depending on the application.In general, there is a trade-off between the averaging property of thekernel and a short response to shocks of the original time series.

[0114] 4.6 Derivative D[τ] and γ-Derivative D[τ,γ]

[0115] The derivative operator $\begin{matrix}{{D\lbrack\tau\rbrack} = \frac{\Delta \lbrack\tau\rbrack}{\tau}} & (34)\end{matrix}$

[0116] behaves exactly as the differential operator Δ[τ], except for thenormalization D[τ; t]=1. This derivative can be iterated in order toconstruct higher order derivatives:

D ² [τ]=D[τ; D[τ]].  (35)

[0117] The range of the second-order derivative operator D² is 2τ. Moregenerally, the n-th order derivative operator D^(n), constructed byiterating the derivative operator n times, has a range nτ. As defined,the derivative operator has the dimension of an inverse time. It iseasier to work with dimensionless operators, and this is done bymeasuring τ in some units. One year provides a convenient unit,corresponding to an annualized return when D[τ]x is computed. The choiceof unit is denoted by τ/1y, meaning that τ is measured in years; otherunits could be taken as well.

[0118] For a random diffusion process, a more meaningful normalizationfor the derivative is to take D[τ]=Δ[τ]/{square root}{square root over(τ/1y)}. For a space of processes as in Section 3.4, such that eq. (9)holds, the basic scaling behavior with τ is eliminated, namelyE[(D[τ]z)²]=σ². More generally, we can define a γ-derivative as$\begin{matrix}{{D\left\lbrack {\tau,\gamma} \right\rbrack} = {\frac{\Delta \lbrack\tau\rbrack}{\left( {{\tau/1}y} \right)^{\gamma}}.}} & (36)\end{matrix}$

[0119] In a preferred embodiment, we use

γ=0 differential,

γ=0.5 stochastic diffusion process,

γ=1 the usual derivative.  (37)

[0120] An empirical probability density function for the derivative isdisplayed in FIG. 8, which plots the annualized derivative D[τ, γ=0.5;x] for USD/CHF from 1 Jan. 1988 to 1 Nov. 1998. The shorter timeintervals τ correspond to the most leptocurtic curves. In order todiscard the daily and weekly seasonality, the computations are done onthe business θ-time scale according to (Dacorogna et al., 1993). Thedata was sampled every 2 hours in θ-time to construct the curves. TheGaussian pdf, added for comparison, has a standard deviation of σ=0.07,similar to that of the other curves. The main part of the scaling with τis removed when the γ-derivative with γ=0.5 is used.

[0121] 4.7 Volatility

[0122] Volatility is a measure widely used for random processes,quantifying the size and intensity of movements, namely the “width” ofthe probability distribution P(Δz) of the process increment ΔZ, where Δis a difference operator yet to be chosen. Often the volatility ofmarket prices is computed, but volatility is a general operator that canbe applied to any time series. There are many ways to turn this ideainto a definition, and there is no unique, universally accepteddefinition of volatility in finance. The most common computation is thevolatility of daily prices, Volatility [x], evaluated for a regular timeseries in business time, with a point-wise price differencer_(l)=Δx_(l)=x(t_(l))−x(t_(l)−τ′) and τ′=1 day. The time horizon τ′ ofthe return is one parameter of the volatility; a second parameter is thelength τ of the moving sample used to compute the “width.” The mostcommon definition for the width estimator uses an L² norm:$\begin{matrix}{{{{Volatility}\left\lbrack {\tau,{\tau^{\prime};z}} \right\rbrack} = \left( {\frac{1}{n}{\sum\limits_{i = 0}^{n - 1}\quad \left( {\delta \quad {{RTS}\left\lbrack {\tau^{\prime};z} \right\rbrack}} \right)_{i}^{2}}} \right)^{1/2}},{{{with}\quad \tau} = {n\quad \tau^{\prime}}},} & (38)\end{matrix}$

[0123] where RTS[τ′; z] is an artificial regular time series, spaced byτ′, constructed from the irregular time series z (see Section 5.3). Theoperator δ computes the difference between successive values (seeSection 5.4).

[0124] The above definition suffers from several drawbacks. First, forinhomogeneous time series, a synthetic regular time series must becreated, which involves an interpolation scheme. Second, the differenceis computed with a point-wise difference. This implies some noise in thecase of stochastic data. Third, only some values at regular time pointsare used. Information from other points of the series, between theregular sampling points, is thrown away. Because of this informationloss, the estimator is less accurate than it could be. Fourth, it isbased on a rectangular weighting kernel (all points have constantweights of either 1 /n or 0 as soon as they are excluded from thesample). A continuous kernel with declining weights leads to a better,less disruptive, and less noisy behavior. Finally, by squaring thereturns, this definition puts a large weight on large changes of z andtherefore increases the impact of outliers and the tails of P(z). Also,as the fourth moment of the probability distribution of the returnsmight not exist (see Müller U. A., Dacorogna M. M., and Pictet O. V.,Heavy tails in high-frequency financial data, in “A practical guide toheavy tails: Statistical Techniques for Analysing Heavy TailedDistributions,” edited by Robert J. Adler, Raisa E. Feldman and Murad S.Taqqu, published by Birkhauser, Boston 1998) (hereinafter Müller et al.,1998), the volatility of the volatility might not exist either. In otherwords, this estimator is not very robust. There are thus several reasonsto prefer a volatility defined as an L¹ norm: $\begin{matrix}{{{{Volatility}\left\lbrack {\tau,{\tau^{\prime};z}} \right\rbrack} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}\quad {{\Delta \left\lbrack {{RTS}\left\lbrack {\tau^{\prime};z} \right\rbrack} \right\rbrack}_{i}}}}},{{{with}\quad \tau} = {N\quad {\tau^{\prime}.}}}} & (39)\end{matrix}$

[0125] There are again many ways to introduce a better definition forinhomogeneous time series. These definitions are variations of thefollowing one, used in a preferred embodiment:

Volatility [τ, τ′, p; z]=MNorm[τ/2, p; Δ[τ′; z]]  (40)

[0126] where the moving norm MNorm is defined by eq. (32) and thedifferential operator Δ of eq. (33) is used. Let us emphasize that ahomogeneous time series is not needed, and that this definition can becomputed simply and efficiently for high-frequency data because itultimately involves only EMAs. Note the division by 2 in the MNorm ofrange τ/2. This is to attain an equivalent of the definition (38) whichis parametrized by the total size rather than the range of the(rectangular) kernel.

[0127] The variations of definition (40) used in alternate embodimentsinclude, first, replacing the norm MNorm by a moving standard deviationMSD, as defined by eq. (32). This modification subtracts the empiricalsample mean from all observations of Δ[τ′; z]. This is not standard forvolatility computations of prices in finance, but might be a betterchoice for other time series or applications. Empirically, for most datain finance (e.g., FX data), the numerical difference between takingMNorm and MSD is very small. The second variation is replacing thedifferential Δ by a γ-derivative D[τ, γ]. The advantage of using thegamma derivative is to remove the leading τ dependence (for example, bydirectly computing the annualized volatility, independently of τ). Anexample is given by FIG. 9, which shows the annualized volatilitycomputed as MNorm [τ/2; D [τ/32, γ=0.5; x]] with τ=1 h. The norm iscomputed with p=2 and n=8. The plotted volatility has 5 main maximacorresponding to the 5 working days of the example week. The Tuesdaymaximum 910 is higher than the others, due to the stock crash mentionedabove.

[0128] Let us emphasize that the volatility definition (38) depends onthe two time ranges τ and τ′ and, to be unambiguous, both time intervalsmust be given. Yet, for example when talking about a daily volatility,the common terminology is rather ambiguous because only one timeinterval is specified. Usually, the emphasis is put on τ′. A dailyvolatility, for example, measures the average size of daily pricechanges, i.e., τ′=1 day. The averaging time range τ is chosen as amultiple of τ′, of the order τ≧τ′ up to τ=1000 τ′ or more. Largermultiples lead to lower stochastic errors as they average over largesamples, but they are less local and dampen the possible time variationsof the volatility. From empirical studies, one can conclude that goodcompromises are in the range from τ=16τ′ to τ=32τ′.

[0129] On other occasions, for example in risk management, one isinterested in the conditional daily volatility. Given the prices up totoday, we want to produce an estimate or forecast for the size of theprice move from today to tomorrow (i.e., the volatility within a smallsample of only one day). The actual value of this volatility can bemeasured one day later; it has τ=1 day by definition. In order tomeasure this value with an acceptable precision, we may choose adistinctly smaller τ′, perhaps τ′=1 hour. Clearly, when only one timeparameter is given, there is no simple convention to remove theambiguity.

[0130] 4.8 Standardized Time Series {circumflex over (z)}, MovingSkewness and Kurtosis

[0131] From a time series z, we can derive a moving standardized timeseries $\begin{matrix}{{\hat{z}\lbrack\tau\rbrack} = {\frac{z - {{MA}\left\lbrack {\tau;z} \right\rbrack}}{{MSD}\left\lbrack {\tau;z} \right\rbrack}.}} & (41)\end{matrix}$

[0132] In finance, z typically stands for a time series of returnsrather than prices.

[0133] Once a standardized time series {circumflex over (z)}[τ] has beendefined, the definitions for the moving skewness and the moving kurtosisare straightforward: $\begin{matrix}\begin{matrix}{{{{MSkewness}\left\lbrack {\tau_{1},{\tau_{2};z}} \right\rbrack} = \quad {{MA}\left\lbrack {\tau_{1};{\hat{z}\left\lbrack \tau_{2} \right\rbrack}^{3}} \right\rbrack}},} \\{{{MKurtosis}\left\lbrack {\tau_{1},{\tau_{2};z}} \right\rbrack} = \quad {{{MA}\left\lbrack {\tau_{1};{\hat{z}\left\lbrack \tau_{2} \right\rbrack}^{4}} \right\rbrack}.}}\end{matrix} & (42)\end{matrix}$

[0134] The three quantities for our sample week are displayed in FIG.10, which shows plots of the standardized return 1020, moving skewness1040, and moving kurtosis 1060. The returns are computed as r=D[τ=15minutes; x] and standardized with τ₁=τ₂=24 h.

[0135] 4.9 Moving Correlation

[0136] Several definitions of a moving correlation can be constructedfor inhomogeneous time series. Generalizing from the statistics textbookdefinition, we can write two simple definitions:

MCorrelation₁ [τ; y, z]=MA[(y−MA[y])(z−MA[z])]/(MSD[y]MSD[z]).  (43)$\begin{matrix}\begin{matrix}{{{MCorrelation}_{2}\left\lbrack {{\tau;y},z} \right\rbrack} = \quad {{MA}\left\lbrack \frac{\left( {y - {{MA}\lbrack y\rbrack}} \right)\left( {z - {{MA}\lbrack z\rbrack}} \right)}{{{MSD}\lbrack y\rbrack}{{MSD}\lbrack z\rbrack}} \right\rbrack}} \\{{= \quad {{MA}\left\lbrack {\hat{y}\hat{\quad z}} \right\rbrack}},}\end{matrix} & (44)\end{matrix}$

[0137] where all the MA and MSD operators on the right hand sides aretaken with the same decay constant τ. These definitions are notequivalent because the MSD operators in the denominator are time seriesthat do not commute with the MA operators. Yet both definitions havetheir respective advantages. The first definition obeys the inequality−1≦MCorrelation₁≦1. This can be proven by noting that MA[z²](t) for agiven t provides a norm on the space of (finite) time series up to t.This happens because the MA operator has a strictly positive kernel thatacts as a metric on the space of time series. In this space, thetriangle inequality holds: {square root}{square root over(MA[(y+z)²])}≦{square root}{square root over (MA[y²])}+{squareroot}{square root over (MA[z²])}, and, by a standard argument, theinequality on the correlation follows. With the second definition (44),the correlation matrix is bilinear for the standardized time series.Therefore, the rotation that diagonalizes the correlation matrix actslinearly in the space of standardized time series. This property isnecessary for multivariate analysis, when a principal componentdecomposition is used.

[0138] In risk management, the correlation of two time series ofreturns, x and y, is usually computed without subtracting the samplemeans of x and y. This implies a variation of eqs. (43) and (44):

MCorrelation₁′[τ; y, z]=MA[(y z]/(MNorm[y]MNorm[z]),  (45)$\begin{matrix}{{{MCorrelation}_{2}^{\prime}\left\lbrack {{\tau;y},z} \right\rbrack} = {{MA}\left\lbrack \frac{yz}{\left( {{{MNorm}\lbrack y\rbrack}{{MNorm}\lbrack z\rbrack}} \right.} \right\rbrack}} & (46)\end{matrix}$

[0139] where again the same τ is chosen for all MA operators.

[0140] In general, any reasonable definition of a moving correlationmust obey

lim MCorrelation[τ; y, z]→p[y,z]  (47)

[0141] where p[y, z] is the theoretical correlation of the twostationary processes x and y. Generalizing the definition (44), therequirements for the correlation kernel are to construct a causal, timetranslation invariant, and a linear operator for y and z . This leads tothe most general representation $\begin{matrix}{{{{MCorrelation}\left\lbrack {\hat{y},\hat{z}} \right\rbrack}(t)} = {\int_{0}^{\infty}{\int_{0}^{\infty}\quad {{t^{\prime}}{t^{''}}{c\left( {t^{\prime},t^{''}} \right)}{\hat{y}\left( {t - t^{\prime}} \right)}{{\hat{z}\left( {t - t^{''}} \right)}.}}}}} & (48)\end{matrix}$

[0142] We also require symmetry between the arguments:MCorrelation[{circumflex over (z)}, ŷ]=MCorrelation[ŷ, {circumflex over(z)}]. Moreover, the correlation must be a generalized average, namelyMCorrelation[Const, Const′]=ConstConst′, or for the kernel∫∫₀^(∞)  t^(′)t^(″)c(t^(′), t^(″)) = 1.

[0143] There is a large selection of possible kernels that obey theabove requirements. For example, eq. (44) is equivalent to the kernel${{c\left( {t^{\prime},t^{''}} \right)} = {{\delta \left( {t^{\prime} - t^{''}} \right)}{ma}\frac{\left( {t^{\prime} + t^{''}} \right)}{2}}},$

[0144] but other choices might be better than this one.

[0145] 4.10 Windowed Fourier Transform

[0146] In order to study a time series and its volatility at differenttime scales, we want to have a method similar to wavelet transformmethods, yet adapted to certain frequencies. As with wavelet transforms,a double representation in time and frequency is needed, but aninvertible transformation is not needed here because our aim is toanalyze rather than further process the signal. This gives us moreflexibility in the choice of the transformations.

[0147] A single causal kernel with the desired properties is or issimilar to ma[τ](t) sin(kt/τ). Essentially, the sine part is (locally)analyzing the signal at a frequency k/τ and the ma part is taking acausal window of range τ. In order to obtain a couple of oscillations inthe window 2τ, choose k between k˜π and k˜5π. Larger k values increasethe frequency resolution at the cost of the time resolution. The basicidea is to compute an EMA with a complex τ; this is equivalent toincluding a sine and cosine part in the kernel. The advantageouscomputational iterative property of the moving average is preserved.

[0148] The first step is to use complex iterated EMAs. The kernel of thecomplex ema is defined as $\begin{matrix}{{{{{ema}\lbrack\zeta\rbrack}(t)} = \frac{^{{- \zeta}\quad t}}{\tau}},{{{where}\quad \zeta} = {\frac{1}{\tau}\left( {1 + {ik}} \right)}},} & (49)\end{matrix}$

[0149] and where ξ is complex (ξ∈C) but τ is again a real number. Thechoice of the normalization factor 1/τ is somewhat arbitrary (a factor|ξ| will produce the same normalization for the real case k=0) but leadsto a convenient definition of the windowed Fourier kernel below. Byusing the convolution formula, one can prove iteratively that the kernelof the complex EMA[ξ, n] is given by $\begin{matrix}{{{e\quad m\quad {a\left\lbrack {\zeta,n} \right\rbrack}(t)} = {\frac{1}{\left( {n - 1} \right)!}\left( \frac{t}{\tau} \right)^{n - 1}\frac{^{{- \zeta}\quad t}}{\tau}}},} & (50)\end{matrix}$

[0150] which is analogous to eq. (26). The normalization is such that,for a constant function c(t)=c, $\begin{matrix}{{E\quad M\quad {A\left\lbrack {\zeta,{n;c}} \right\rbrack}} = {\frac{c}{\left( {1 + {i\quad k}} \right)^{n}}.}} & (51)\end{matrix}$

[0151] Using techniques similar to those applied to eq. (23), we obtainan iterative computational formula for the complex EMA: $\begin{matrix}\begin{matrix}{{{E\quad M\quad {A\left\lbrack {\zeta;z} \right\rbrack}\left( t_{n} \right)} = \quad {{\mu \quad E\quad M\quad {A\left\lbrack {\zeta;z} \right\rbrack}\left( t_{n - 1} \right)} + {z_{n - 1}v} - \frac{\mu}{1 + {i\quad k}} + {z_{n}1} - \frac{v}{1 + {i\quad k}}}},{with}} \\{\alpha = \quad {\zeta \left( {t_{n} - t_{n - 1}} \right)}} \\{\mu = \quad ^{- \alpha}}\end{matrix} & (52)\end{matrix}$

[0152] where v depends on the chosen interpolation scheme as given byeq. (24).

[0153] We define the (complex) kernel wf(t) of the windowed Fouriertransform WF as $\begin{matrix}\begin{matrix}{{w\quad {f\left\lbrack {\tau,k,n} \right\rbrack}(t)} = \quad {m\quad {a\left\lbrack {\tau,n} \right\rbrack}(t)^{\quad k\quad {t/\tau}}}} \\{= \quad {\frac{1}{n}{\sum\limits_{j = 1}^{n}{\frac{1}{\left( {j - 1} \right)!}\left( \frac{t}{\tau} \right)^{j - 1}\frac{^{{- \zeta}\quad t}}{\tau}}}}} \\{= \quad {\frac{1}{n}{\sum\limits_{j = 1}^{n}{e\quad m\quad {a\left\lbrack {\zeta,j} \right\rbrack}{(t).}}}}}\end{matrix} & (53)\end{matrix}$

[0154]FIG. 11 plots the kernel wf(t) for the windowed Fourier operatorWF, for n=8 and k=6. Three aspects of the complex kernel are shown: (1)the envelope 1120 (=absolute value), (2) the real part 1140 (starting ontop), and (3) the imaginary part 1160 (starting at zero). Anotherappropriate name for this operator might be CMA for Complex MovingAverage. The normalization is such that, for a constant function c(t)=c,$N_{W\quad F} = {{W\quad {F\left\lbrack {\zeta,{n;c}} \right\rbrack}} = {\frac{c}{n}{\sum\limits_{j = 1}^{n}{\frac{1}{\left( {1 + {i\quad k}} \right)^{j}}.}}}}$

[0155] In order to provide a more convenient real quantity, with themean of the signal subtracted, we can define a (non-linear) normedwindowed Fourier transform as

NormedWF[ξ, n; z]=|WF[ξ, n; z]−N_(WF)MA[τ, n; z]|  (54)

[0156] The normalization is chosen so that

NormaedWF[ξ, n; c]=0

[0157] Note that in eq. (54) we are only interested in the amplitude ofthe measured frequency; by taking the absolute value we have lostinformation on the phase of the oscillations.

[0158]FIG. 12 shows an example of the normed windowed Fourier transformfor the example week, wherein the normed windowed Fourier transform isplotted, with τ=1 hour, k=6, and n=8. The stock market crash is againnicely spotted as the peak 1220 on Tuesday 28.

[0159] Using the described methods, other quantities of interest can beeasily calculated. For example, we can compute the relative share of acertain frequency in the total volatility. This would mean a volatilitycorrection of the normed windowed Fourier transform. A way to achievethis is to divide NormedWF by a suitable volatility, or to replace z bythe standardized time series {circumflex over (z)} in eq. (54).

[0160] 5 Implementation

[0161] In a preferred embodiment, the techniques described above areimplemented in a method used to obtain predictive information forinhomogeneous financial time series. Major steps of the method (see FIG.13) comprise the following: At step 1310 financial market transactiondata is electronically received by a computer over an electronicnetwork. At step 1320 the received financial market transaction data iselectronically stored in a computer-readable medium accessible to thecomputer (e.g., on a hard drive, in RAM, or on an optical storage disk).At step 1330 a time series z is constructed that models the receivedfinancial market transaction data. At step 1340 an exponential movingaverage operator is constructed, and at step 1350 an iteratedexponential moving average operator is constructed that is based on theexponential moving operator constructed in step 1340. At step 1360, atime-translation-invariant, causal operator Ω[z] is constructed that isbased on the iterated exponential moving average operator constructed instep 1350. Ω[z] is a convolution operator with kernel ω and time rangeτ. (It is important to note, with respect to the steps described herein,that no particular order, other than that order required to make themethod practical, should be inferred from the fact that the steps aredescribed and labeled in a certain order; the order has been chosenmerely for ease of explication.) At step 1370 values of one or morepredictive factors relating to said time series z are calculated by thecomputer. The predictive factors are defined in terms of the convolutionoperator Ω[z]. At step 1380 the values of one or more predictive factorscalculated by the computer are stored in a computer readable medium (notnecessarily the same medium as in step 1320).

[0162] Various predictive factors have been described above, andspecifically comprise return, momentum, and volatility. Other predictivefactors will be apparent to those skilled in the art.

[0163] In a second preferred embodiment, the major steps of the methoddiffer from those described in connection with FIG. 13. This secondpreferred embodiment is illustrated in FIG. 14. Major steps of themethod comprise the following: At step 1410 financial market transactiondata is electronically received by a computer over an electronicnetwork. At step 1420 the received financial market transaction data iselectronically stored in a computer-readable medium accessible to thecomputer. At step 1430 a time series z is constructed that models thereceived financial market transaction data. At step 1440 an exponentialmoving operator is constructed, and at step 1450, an iteratedexponential moving operator is constructed that is based on theexponential moving operator constructed in step 1440. At step 1460, atime-translation-invariant, causal operator Ω[z] is constructed that isbased on the iterated exponential moving average operator constructed instep 1450. Ω[z] is a convolution operator with kernel ω and time rangeτ. At step 1470 a standardized time series {circumflex over (z)} isconstructed (see Section 4.8). At step 1480 values of one or morepredictive factors relating to said time series z are calculated by thecomputer. The predictive factors in this case are defined in terms ofthe standardized time series {circumflex over (z)}. At step 1490 thevalues of one or more predictive factors calculated by the computer arestored in a computer readable medium (not necessarily the same medium asin step 1420). In addition to the predictive factors mentioned above,additional predictive factors relevant to this method are movingskewness and moving kurtosis.

[0164] In a third preferred embodiment, the major steps are similar tothose described above, but differ enough to call for separateexplanation. In this embodiment, illustrated in FIG. 15, the steps areas follows: At step 1510 financial market transaction data iselectronically received by a computer over an electronic network. Atstep 1520 the received financial market transaction data iselectronically stored in a computer-readable medium accessible to thecomputer. At step 1530 a time series z is constructed that models thereceived financial market transaction data. At step 1540 an exponentialmoving average operator is constructed, and at step 1550 an iteratedexponential moving average operator is constructed that is based on theexponential moving operator constructed in step 1540. At step 1555, atime-translation-invariant, causal operator Ω[z] is constructed that isbased on the iterated exponential moving average operator constructed instep 1550. Ω[z] is a convolution operator with kernel ω and time rangeτ. At step 1550 an exponential moving average operator EMA[τ; z] isconstructed, and at step 1560 a moving average operator MA isconstructed. MA depends on EMA (see Section 4.3). At step 1565 a movingstandard deviation operator MSD is constructed. MSD depends on MA (seeSection 4.4). At step 1570 values of one or more predictive factorsrelating to said time series z are calculated by the computer. Thepredictive factors are defined in terms of one or more of EMA, MA, andMSD. At step 1580 the values of one or more predictive factorscalculated by the computer are stored in a computer readable medium (notnecessarily the same medium as in step 1520).

[0165] In a fourth preferred embodiment of the invention, the majorsteps (illustrated in FIG. 16) are again similar to those describedabove, but merit separate description. At step 1610 financial markettransaction data is electronically received by a computer over anelectronic network. At step 1620 the received financial markettransaction data is electronically stored in a computer-readable mediumaccessible to the computer. At step 1630 a time series z is constructedthat models the received financial market transaction data. At step 1640a complex iterated exponential moving average operator EMA[τ; z], withkernel ema, is constructed (see Section 4.10). At step 1650 atime-translation-invariant, causal operator Ω[z] is constructed. Ω[z] isa convolution operator with kernel ω and time range τ, and is based onthe operator constructed in step 1640. At step 1660 a windowed Fouriertransform WF is constructed (WF is defined in terms of EMA; see Section4.10). At step 1670 values of one or more predictive factors relating tosaid time series z are calculated by the computer. The predictivefactors are defined in terms of the windowed Fourier transform WF. Atstep 1680 the values of one or more predictive factors calculated by thecomputer are stored in a computer readable medium (not necessarily thesame medium as in step 1620).

[0166] Although the subject invention has been described with referenceto preferred embodiments, numerous modifications and variations can bemade that will still be within the scope of the invention. No limitationwith respect to the specific embodiments disclosed herein is intended orshould be inferred.

What is claimed is:
 1. A method of obtaining predictive information forinhomogeneous financial time series, comprising the steps of:electronically receiving financial market transaction data over anelectronic network; electronically storing in a computer-readable mediumsaid received financial market transaction data; constructing aninhomogeneous time series z that represents said received financialmarket transaction data; constructing an exponential moving averageoperator; constructing an iterated exponential moving average operatorbased on said exponential moving average operator; constructing atime-translation-invariant, causal operator Ω[z] that is a convolutionoperator with kernel ω and that is based on said iterated exponentialmoving average operator; electronically calculating values of one ormore predictive factors relating to said time series z, wherein said oneor more predictive factors are defined in terms of said operator Ω[z];and electronically storing in a computer readable medium said calculatedvalues of one or more predictive factors.
 2. The method of claim 1,wherein said operator Ω[z] has the form: $\begin{matrix}{{{\Omega \lbrack z\rbrack}(t)} = \quad {\int_{- \infty}^{t}\quad {{t^{\prime}}{\omega \left( {t - t^{\prime}} \right)}{z\left( t^{\prime} \right)}}}} \\{= \quad {\int_{0}^{\infty}\quad {{t^{\prime}}{\omega \left( t^{\prime} \right)}{{z\left( {t - t^{\prime}} \right)}.}}}}\end{matrix}$


3. The method of claim 1, wherein said exponential moving averageoperator EMA[τ; z] has the form: $\begin{matrix}\begin{matrix}{{{E\quad M\quad {A\left\lbrack {\tau;z} \right\rbrack}\left( t_{n} \right)} = \quad {{\mu \quad E\quad M\quad \left. {A\left( {\tau;z} \right.} \right\rbrack \left( t_{n - 1} \right)} + {\left( {v - \mu} \right)z_{n - 1}} + {\left( {1 - v} \right)z_{n}}}},{with}} \\{{\alpha = \quad \frac{\tau}{t_{n} - t_{n - 1}}},} \\{{\mu = \quad ^{- \alpha}},}\end{matrix} & (23)\end{matrix}$

where v depends on a chosen interpolation scheme.
 4. The method of claim1, wherein said operator Ω[z] is a differential operator Δ[τ] that hasthe form: Δ[τ]=γ(EMA[ατ, 1]+EMA[ατ, 2]−2EMA[αβτ, 4]), where γ is fixedso that the integral of the kernel of the differential operator from theorigin to the first zero is 1; α is fixed by a normalization conditionthat requires Δ[τ; c]=0 for a constant c; and β is chosen in order toget a short tail for the kernel of the differential operator Δ[τ]. 5.The method of claim 4 wherein said one or more predictive factorscomprises a return of the form r[τ]=Δ[τ; x], where x represents alogarithmic price.
 6. The method of claim 1 wherein said one or morepredictive factors comprises a momentum of the form x−EMA[τ; x], where xrepresents a logarithmic price.
 7. The method of claim 1 wherein saidone or more predictive factors comprises a volatility.
 8. The method ofclaim 7 wherein said volatility is of the form: Volatility[τ, τ′, p;z]=MNorm[τ/2,p; Δ[τ′; z]], where MNorm[τ, p; z]=MA[τ; |z|^(p)]^(1/p),and${{M\quad {A\left\lbrack {\tau,n} \right\rbrack}} = {\frac{1}{n}{\sum\limits_{k = 1}^{n}{E\quad M\quad {A\left\lbrack {\tau^{\prime},k} \right\rbrack}}}}},{{{with}\quad \tau^{\prime}} = \frac{2\quad \tau}{n + 1}},$

and where p satisfies 0<p≦2, and τ′ is a time horizon of a returnr[τ]=Δ[τ; x], where x represents a logarithmic price.
 9. A method ofobtaining predictive information for inhomogeneous financial timeseries, comprising the steps of: electronically receiving financialmarket transaction data over an electronic network; electronicallystoring in a computer readable medium said received financial markettransaction data; constructing an inhomogeneous time series z thatcorresponds to said received financial market transaction data;constructing an exponential moving average operator; constructing aniterated exponential moving average operator based on said exponentialmoving average operator; constructing a time-translation-invariant,causal operator Ω[z] that is a convolution operator with kernel ω andthat is based on said iterated exponential moving average operator;constructing a standardized time series {circumflex over (z)};electronically calculating values of one or more predictive factorsrelating to said time series z, wherein said one or more predictivefactors are defined in terms of said standardized time series{circumflex over (z)}; and electronically storing in a computer readablemedium said calculated values of one or more predictive factors.
 10. Themethod of claim 9 wherein the standardized time series {circumflex over(z)} is of the form:${{\hat{z}\lbrack\tau\rbrack} = \frac{z - {M\quad {A\left\lbrack {\tau;z} \right\rbrack}}}{M\quad S\quad {D\left\lbrack {\tau;z} \right\rbrack}}},$

where${{{MA}\left\lbrack {\tau,n} \right\rbrack} = {\frac{1}{n}{\sum\limits_{k = 1}^{n}\quad {{EMA}\left\lbrack {\tau^{\prime},k} \right\rbrack}}}},{{{with}\quad \tau^{\prime}} = \frac{2\tau}{n + 1}},{and}$

where MSD[τ, p; z]=MA[τ;|z−MA[τ; z]|^(p)]^(1/p).
 11. The method of claim9 wherein said one or more predictive factors comprises a movingskewness.
 12. The method of claim 11 wherein said moving skewness is ofthe form: MSkewness[τ₁, τ₂; z]=MA[τ₁; {circumflex over (z)}[τ₂ where τ₁is the length of a time interval around time “now” and τ₂ is the lengthof a time interval around time “now−τ”.
 13. The method of claim 12wherein the standardized time series z is of the form:${{\hat{z}\lbrack\tau\rbrack} = \frac{z - {{MA}\left\lbrack {\tau;z} \right\rbrack}}{{MSD}\left\lbrack {\tau;z} \right\rbrack}},$

where${{{MA}\left\lbrack {\tau,n} \right\rbrack} = {\frac{1}{n}{\sum\limits_{k = 1}^{n}\quad {{EMA}\left\lbrack {\tau^{\prime},k} \right\rbrack}}}},{{{with}\quad \tau^{\prime}} = \frac{2\tau}{n + 1}},$

and where MSD[τ, p; z]=MA[τ;|z−MA[τ; z]|^(p)]^(1/p).
 14. The method ofclaim 9 wherein said one or more predictive factors comprises a movingkurtosis.
 15. The method of claim 14 wherein said moving kurtosis is ofthe form MKurtosis[τ₁, τ₂; z]=MA[τ₁; {circumflex over (z)}[τ₂]⁴], whereτ₁ is the length of a time interval around time “now” and τ₂ is thelength of a time interval around time “now−τ.”
 16. The method of claim15 wherein the standardized time series {circumflex over (z)} is of theform:${{\hat{z}\lbrack\tau\rbrack} = \frac{z - {{MA}\left\lbrack {\tau;z} \right\rbrack}}{{MSD}\left\lbrack {\tau;z} \right\rbrack}},$

where${{{MA}\left\lbrack {\tau,n} \right\rbrack} = {\frac{1}{n}{\sum\limits_{k = 1}^{n}\quad {{EMA}\left\lbrack {\tau^{\prime},k} \right\rbrack}}}},{{{with}\quad \tau^{\prime}} = \frac{2\tau}{n + 1}},$

and where MSD[τ, p; z]=MA[τ;|z−MA[τ; z]|^(p)]^(1/p).
 17. A method ofobtaining predictive information for inhomogeneous financial timeseries, comprising the steps of: electronically receiving financialmarket transaction data over an electronic network; electronicallystoring in a computer readable medium said received financial markettransaction data; constructing an inhomogeneous time series z thatcorresponds to said received financial market transaction data;constructing an exponential moving average operator EMA[τ; z];constructing an iterated exponential moving average operator based onsaid exponential moving average operator EMA[τ; z]; constructing atime-translation-invariant, causal operator Ω[z] that is a convolutionoperator with kernel ω and time range τ, and that is based on saiditerated exponential moving average operator; constructing a movingaverage operator MA that depends on said EMA operator; constructing amoving standard deviation operator MSD that depends on said MA operator;electronically calculating values of one or more predictive factorsrelating to said time series z, wherein said one or more predictivefactors depend on one or more of said operators EMA, MA, and MSD; andelectronically storing in a computer readable medium said calculatedvalues of one or more predictive factors.
 18. The method of claim 17wherein said one or more predictive factors comprises a movingcorrelation.
 19. The method of claim 18 wherein said moving correlationis of the form:MCorrelation[ŷ, ẑ](t) = ∫₀^(∞)∫₀^(∞)  t^(′)t^(″)c(t^(′), t^(″))ŷ(t − t^(′))ẑ(t − t^(″)).

.
 20. A method of obtaining predictive information for inhomogeneousfinancial time series, comprising the steps of: electronically receivingfinancial market transaction data over an electronic network;electronically storing in a computer readable medium said receivedfinancial market transaction data; constructing an inhomogeneous timeseries z that corresponds to said received financial market transactiondata; constructing a complex iterated exponential moving averageoperator EMA[τ; z], with kernel ema; constructing atime-translation-invariant, causal operator Ω[z] that is a convolutionoperator with kernel ω and time range τ, and that is based on saidcomplex iterated exponential moving average operator; constructing awindowed Fourier transform WF that depends on said EMA operator;electronically calculating values of one or more predictive factorsrelating to said time series z, wherein said one or more predictivefactors depend on said windowed Fourier transform; and electronicallystoring in a computer readable medium said calculated values of one ormore predictive factors.
 21. The method of claim 20 wherein said complexiterated exponential moving average operator EMA has a kernel ema of theform:${{{{ema}\left\lbrack {\zeta,n} \right\rbrack}(t)} = {\frac{1}{\left( {n - 1} \right)!}\left( \frac{t}{\tau} \right)^{n - 1}\frac{^{{- \zeta}\quad t}}{\tau}}},$

where ξ∈C, with$\quad {\zeta = {\frac{1}{\tau}{\left( {1 + {ik}} \right).}}}$


22. The method of claim 20 wherein EMA is computed using the iterativecomputational formula: $\begin{matrix}{{{{{EMA}\left\lbrack {\zeta;z} \right\rbrack}\left( t_{n} \right)} = {{\mu \quad {{EMA}\left\lbrack {\zeta;z} \right\rbrack}\left( t_{n - 1} \right)} + {z_{n - 1}\frac{\nu - \mu}{1 + {ik}}} + {z_{n}\frac{1 - \nu}{1 + {ik}}}}},{with}} \\{\alpha = {\zeta \left( {t_{n} - t_{n - 1}} \right)}} \\{\mu = ^{- \alpha}}\end{matrix}$

where v depends on a chosen interpolation scheme.
 23. The method ofclaim 20 wherein said windowed Fourier transform has a kernel wf of theform:${{{wf}\left\lbrack {\tau,k,n} \right\rbrack}(t)} = {\frac{1}{n}{\sum\limits_{j = 1}^{n}\quad {{{ema}\left\lbrack {\zeta,j} \right\rbrack}{(t).}}}}$


24. The method of claim 23 wherein said ema is of the form:${{{{ema}\left\lbrack {\zeta,n} \right\rbrack}(t)} = {\frac{1}{\left( {n - 1} \right)!}\left( \frac{t}{\tau} \right)^{n - 1}\frac{e^{{- \zeta}\quad t}}{\tau}}},$

where ξ∈C, with $\zeta = {\frac{1}{\tau}{\left( {1 + {ik}} \right).}}$


25. A method of obtaining predictive information for inhomogeneous timeseries, comprising the steps of: electronically receiving time seriesdata over an electronic network; electronically storing in acomputer-readable medium said received time series data; constructing aninhomogeneous time series z that represents said time series data;constructing an exponential moving average operator; constructing aniterated exponential moving average operator based on said exponentialmoving average operator; constructing a time-translation-invariant,causal operator Ω[z] that is a convolution operator with kernel ω andthat is based on said iterated exponential moving average operator;electronically calculating values of one or more predictive factorsrelating to said time series z, wherein said one or more predictivefactors are defined in terms of said operator Ω[z]; and electronicallystoring in a computer readable medium said calculated values of one ormore predictive factors.